Deterministic homogenization of elliptic equations with lower order terms

For a class of linear elliptic equations of general type with rapidly oscillating coefficients, we use the sigma-convergence method to prove the homogenization result and a corrector-type result. In the case of asymptotic periodic coefficients we derive the optimal convergence rates for the zero order approximation of the solution with no smoothness on the coefficients, in contrast to what has been done up to now in the literature. This follows as a result of the existence of asymptotic periodic correctors for general nonsmooth coefficients. The homogenization process is achieved through a compactness result obtained by proving a Helmholtz-type decomposition theorem in case of Besicovitch spaces.